Higher dimensional rewriting aims to deduce homotopical and homological properties of objects from their presentation. Squier's Theorem for example, asserts that if a monoid admits a finite convergent presentation, then it satisfies some homotopical and homological finiteness conditions. Since then this result has been improved, and extended to other structures such as associative algebras, PROs or PROPs. However, these results suffer from a number of shortcomings: although the methods are constructive, explicit computations in higher dimensions quickly become intractable. Additionally, a general framework unifying the results on all of these structures is still lacking.
In this talk, we investigate a variation on these constructions. First, we embed monoids into so-called "Gray monoids": monoid objects in strict omega-groupoids. This is analogous to seeing associative algebras as (positively graded) dg-algebras concentrated in degree zero. Gray monoids come equipped with a Quillen model structure inherited from that of strict omega-groupoids. Starting from a suitable presentation of a monoid M, we show how to compute a cofibrant replacement of M in this category. The proof has several key improvements over earlier results: some hypotheses are dropped or relaxed, and the construction is more modular, clarifying what does or does not depend on the monoid structure of M.
As a consequence, we believe that this setting is a promising one to explore in order to find a systematic treatment of higher dimensional rewriting.